Large Hermitian hull GRS codes of any given length

The construction of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes of many specific lengths and large dimensions has been an active topic. The construction of Euclidean self-dual GRS codes and twisted generalized Reed-Solomon (TGRS) codes attracts some attentions. In this paper, we construct GRS \([n, k, n-k+1]_{q^2}\) ${[n,k,n-k+1]}_{q^2}$ codes (thus MDS codes) over \(\textbf{F}_{q^2}\) ${\mathbf{F}}_{q^2}$ of the arbitrary length n satisfying \(n \le q^2+1\) $n\le q^2+1$ and any given distance d satisfying \(d=O(q^2)\) $d=O(q^2)$ , such that the dimensions \(h \le k\) $h\le k$ of its Hermitian hull is at least \(h=O(k)\) $h=O(k)$ . This work is a natural extension of previous constructions of Hermitian self-orthogonal GRS codes of many specific lengths. Our method can be used to construct large Hermitian hull MDS TGRS codes of the length \(n|(q^2-1)\) $n|(q^2-1)$ .